30 November, 2012

From differential solution set to Fractal appearance set.


I've given some thought to the recent research published by a team of Scientists that the large scale structure of the Universe looks a lot like the small scale structure of neuronal connections and like the planet scale structure of the connections between computers on the internet.

Here is my explanation for why this is so (yes, this is the type of conversation I prefer to have to the detriment of my dating life) it's simple. From exploration of the class of equations known as differential equations we know that defining the initial conditions of an equation gives rise to a set of solutions for those initial conditions that range often over continuous space. Even if the solutions are not continuous the variation in variables (known as the order of the equation) can be infinite, leading to an open set of possible solutions to the initial equation.

Thus for sufficiently chosen initial conditions any arbitrary solution can be identified for any given equation. It turns out that this is true for those particular equations that have range across continuous space. These tend to have non linear relationships and dynamic variable relationships in dimension (partial)...these type of differential equations are in fact very special...they are the type that are most fitting for solving some of our most complex dynamic systems in the physical world.

Some n'th(where "n" is high not necessarily infinite) order, non linear , partial , differential equations of note:

The Heat equation (how heat energy convects between materials):

The Navier Stokes equation (how fluids [which include gases] flow):

The Schrodinger equation (the governing equation of all quantum mechanics):

The Gravitational Equation (Einstein Field Equation, governs how energy and matter distort space time):

:Looking at those it should be clear why over sufficiently varied initial conditions they seem to give systems of solution that appear similar as the initial conditions vary across wide, near infinite ranges, all vary enough away from any given set of conditions as to appear nearly random....*chaotic* but based on the same kernel the same dna in the form of the initial differential equation and the defined initial conditions. This sounds like self similarity across the solution set and corresponding initial conditions. Fractal systems are a special subclass of Chaotic systems and so from the variations of parameters over infinite sets all those equations appear to create fractal solution sets...which in the aggregate indeed do look alike (not quite random noise...but no real repeatable structures outside of self similar ones). Fractals.

This shouldn't be surprising at all to us, it's not the first time such similarities have been noted but I find it interesting that the connection between the family of solutions that come out of the types of equations that model these systems and how they appear at different scales (essentially different initial condition values) and the similarity across those equations in this because of their identical mathematical form has not been (at least to my knowledge yet made) so I make it here.

09 November, 2012

Simulated Universe, suffers same problem as God.


Consider this:

Assuming that we are in a simulation of the ancestry of beings in another Universe, the question of the finite evolution of the Universe that those beings inhabits still subjects them to what we know about how the Universe is constructed no?

Would they not also be constrained by the forces of nature to design and build their simulation (us) and therefor aren't they also embedded and constrained as well.

Now, let's assume that they are a top level lifeform, meaning they themselves were NOT created as part of another Universes creation event...can we explore (either rigorously or by logic) whether or not this assertion is probable??

Is it possible that they are a top level being and in fact what is the likelihood of top level beings being the creators of our Universe.

I think if we accept the previous hypothesis, that all beings the simulated and the simulators are constrained by the limits of a common physics then we must conclude that if the variation of universal parameters that is supposed for universes in a multiverse is true that there is an infinite number of universal configurations. Not just ones conducive to the running of our "simulation" if that is what we are, or our top level Universe (if that is what we are).

If there is indeed an infinity of Universes...then the subset of Universes which have the conditions necessary to emerge top level beings that would then create Universes is itself infinite but still a smaller Infinity than the whole of the multiverse. If there is an infinite number of multiverses then the percentage of them that are top level Universes is itself *also infinite* leaving it as a certainty that they are NOT simulations.

The question is then, what is the probability that we are a simulation versus a top level Universe given the above.

Again, since the number of Universes with top level beings is infinite and the probability of their creating simulations is finite...then the probability that we are simulation is lower than if we are a top level universe (simply because the number of simulations must necessarily be a subset of the number of top level Universes *even if* that number is infinite).

Note it is no longer important to inquisite further, the answer is in the last line above:

"the probability that we are simulation is lower than if we are a top level universe"

How much lower? Well, since we already assumed that the number of Universes with beings that could even create a simulation is an infinite subset of all possible universes) and we also accepted that the percentage of these that actually created simulations of their ancestry are also an infinite subset...then across all the multiverse...the probability than any selected Universe is spawned via simulation dwindles to zero (since it always remains an infinite subset of Universes which themselves are an infinite subset of all possible Universes...those with and without creative beings).

So by a limit argument I conclude that we are likely NOT a simulation (I used a similar limit argument to conclude that we are also likely not the result of any God a few years ago) it would seem to be true in both cases from this intuitive argument comparing relative infinities no need to go into more rigorous analysis...I could be wrong but I've yet encountered a valid counter argument. Thoughts?